Theorem dcned 2373

Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)

Hypothesis

Ref	Expression
dcned.eq	⊢ (𝜑 → DECID 𝐴 = 𝐵)

Assertion

Ref	Expression
dcned	⊢ (𝜑 → DECID 𝐴 ≠ 𝐵)

Proof of Theorem dcned

Step	Hyp	Ref	Expression
1		dcned.eq	. . 3 ⊢ (𝜑 → DECID 𝐴 = 𝐵)
2		dcn 843	. . 3 ⊢ (DECID 𝐴 = 𝐵 → DECID ¬ 𝐴 = 𝐵)
3	1, 2	syl 14	. 2 ⊢ (𝜑 → DECID ¬ 𝐴 = 𝐵)
4		df-ne 2368	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
5	4	dcbii 841	. 2 ⊢ (DECID 𝐴 ≠ 𝐵 ↔ DECID ¬ 𝐴 = 𝐵)
6	3, 5	sylibr 134	1 ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 835   = wceq 1364   ≠ wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836  df-ne 2368

This theorem is referenced by:  nn0n0n1ge2b  9405  algcvgblem  12217